Would that option not also exist for a player that was drafted, but did not sign by the deadline? Isn't the process over, thereby allowing a player the freedom to go to MLB tryouts, or sign later, before the next draft?
That is not allowed. It would be a tremendous advantage for the Big Money teams and create problems with the draft. If it were allowed, Big Market teams would be signing other teams draft picks after the deadline.
Draft eligible player who go "undrafted" can sign as free agents. Drafted players have to sign with the team that drafted them or wait another year so long as the club does everything correctly.
Some people might not know that high school seniors that go undrafted are actually free agents for the summer. Matt Wieters went undrafted out of high school and was offered free agent deals during the summer. Buster Posey was drafted in the 50th round and that eliminated him from signing as a FA.